Abstract: This paper addresses a study of the eventual regularity of a wave
equation with boundary dissipation and distributed damping. The
equation under consideration is rewritten as a system of first order
and analyzed by semigroup methods. By a certain asymptotic expansion
theorem, we prove that the associated solution semigroup is
eventually differentiable. This implies the eventual regularity of
the solution of the wave equation.

Abstract: An abstract $\nu$-metric was introduced in [1], with a
view towards extending the classical $\nu$-metric of Vinnicombe from
the case of rational transfer functions to more general nonrational
transfer function classes of infinite-dimensional linear control
systems. Here we give an important concrete special
instance of the abstract $\nu$-metric, namely the case when the
ring of stable transfer functions is the Hardy algebra $H^\infty$,
by verifying that all the assumptions demanded in the abstract
set-up are satisfied. This settles the open question implicit
in [2].

Abstract: We consider the
Euler-Bernoulli equation coupled with a wave equation in a bounded
domain. The Euler-Bernoulli has clamped boundary conditions and the
wave equation has Dirichlet boundary conditions. The damping which
is distributed everywhere in the domain under consideration acts
through one of the equations only; its effect is transmitted to the
other equation through the coupling. First we consider the case
where the dissipation acts through the Euler-Bernoulli equation. We
show that in this case the coupled system is not exponentially
stable. Next, using a frequency domain approach combined with the
multiplier techniques, and a recent result of Borichev and Tomilov
on polynomial decay characterization of bounded semigroups, we
provide precise decay estimates showing that the energy of this
coupled system decays polynomially as the time variable goes to
infinity. Second, we discuss the case where the damping acts through
the wave equation. Proceeding as in the first case, we prove that
this new system is not exponentially stable, and we provide precise
polynomial decay estimates for its energy. The results obtained
complement those existing in the literature involving the hinged
Euler-Bernoulli equation.

Abstract: This paper deals with the Pontryagin's principle of optimal control
problems governed by the 2D Navier-Stokes equations with integral state
constraints and coupled integral control--state
constraints. As an application, the necessary conditions for
the local solution in the sense of $L^r(0,T;L^2(\Omega))$ ($2 < r < \infty$)
are also obtained.

Abstract: Momentum (or trend-following) trading strategies are widely used in the
investment world. To better understand the nature of trend-following trading
strategies and discover the corresponding optimality conditions,
we consider the cases when the market trends are fully observable.
In this paper, the market follows a regime switching model with
three states (bull, sideways, and bear).
Under this model, a set of sufficient
conditions are developed to guarantee the optimality of
trend-following trading strategies.
A dynamic programming approach is used to verify these
optimality conditions. The value functions are
characterized by the associated HJB equations and are shown to be
either linear functions or infinity depending on the parameter
values.
The results in this paper will help an investor to identify market conditions
and to avoid trades which might be unprofitable even under the best market
information.
Finally, the corresponding value functions will provide
an upper bound for trading performance which can be used as a
general guide to rule out unrealistic expectations.